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Introduction to Mesoscale Methods

Ronojoy Adhikari
September 09, 2009

Introduction to Mesoscale Methods

Lecture at the Understanding Molecular Simulations school at the Jawaharlal Nehru Center for Advanced Scientific Research.

Ronojoy Adhikari

September 09, 2009
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  1. Mesoscale Simulation Methods
    Ronojoy Adhikari
    The Institute of Mathematical Sciences
    Chennai

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  2. Outline
    • What is mesoscale ?
    • Mesoscale statics and dynamics through coarse-graining.
    • Coarse-grained equations for a binary fluid-fluid mixture.
    • Numerical methods of solution.
    • Example simulations.

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  3. Numbers and methods
    1
    2
    3
    4
    5
    6
    7
    0
    Ab initio methods
    Atomistic methods
    Continuum methods
    DFT
    CPMD
    MD
    LD
    BD
    DPD
    CFD
    LB
    Number of atoms (log) Simulation method Example of method

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  4. How lengths scale with numbers
    !! Try this for water
    Molar mass = 18 gm
    Molar volume = 18 mL
    N = 1023
    L 107˚
    A
    L N 1
    3
    N = 1
    L 1˚
    A
    !! Are other scalings possible ? How does length grow with
    size for a polymer ?
    Throughout, (!!) indicate exercises/derivations/points to ponder.

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  5. Length and time scales
    pico
    milli
    micro
    nano
    nano micro milli
    L
    T
    H = E
    m ¨
    R = ⇥U
    ˙
    R = U
    ˙
    c = D 2c
    !! Are there systems with millimeter L but picosecond T ?

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  6. How times scale with masses and lengths

    x + kx = 0
    2
    0
    = k
    m
    0
    m
    k
    ¨
    u = c2 2u
    0
    = ±cq
    ⇥0 c
    ˙
    u = D 2u
    i 0
    = Dq2
    ⇥0
    2
    D
    Harmonic Oscillator Wave Equation Diffusion Equation

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  7. Thermal fluctuations
    P(x |x) =
    1

    2 D⇥
    exp
    (x x)2
    2D⇥

    Brownian motion : Einstein (1905)
    D = kBT
    6⇥ a
    temperature
    size
    Stokes-Einstein-Sutherland Relation
    Q. Why does a smaller particle have a higher
    diffusion coefficient ?
    A. The ‘root N’ effect! The central limit theorem
    helps us understand this.
    !! Is it (central limit) theorem or
    central (limit theorem) ?

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  8. Langevin theory
    inertia
    damping reversible force
    fluctuation
    deterministic stochastic
    mean behaviour ‘root N’ fluctuations
    regression to the mean
    equilibrium fluctuations
    v2⇥ = kBT
    F = 0 Free Brownian particle

    v + v F(x) = ⇥(t)

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  9. Mesoscale regime
    1
    2
    3
    4
    5
    6
    7
    0
    Ab initio
    Atomistic
    Continuum
    Number of atoms Simulation
    Coarse grained length scales.
    Coarse grained time scales.
    Retain thermal fluctuations.
    Mesoscale methods
    }
    Examples
    Brownian dynamics.
    Dissipative particle dynamics.
    Time-dependent Ginzburg-Landau.

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  10. Coarse-graining in degrees of freedom
    What is the idea ?
    x1
    x2
    } P(x1, x2
    )
    RV y = x1
    + x2
    “Coarse-grained” sum
    P(y)
    P(y) = dx1dx2
    (y x1 x2
    )P(x1, x2
    )
    What is the distribution ?
    contains less information about the system than
    P(y) P(x1, x2
    )
    Adequate if we are only interested in the sum variable.

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  11. General ‘coarse-graining’ formula
    P(x1, x2, . . . , xN
    ) y = f(x1, x2, . . . , xN
    )
    Microstate probability Mesoscale variable/order parameter
    P(y) =
    ⇥ N
    i=1
    dxi
    [y f(x1, . . . , xN
    )]P(x1, . . . , xN
    )
    Mesostate probability

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  12. Coarse-grained Landau-Ginzburg functional
    Microscopic Hamiltonian Gibbs distribution
    P(q) =
    exp[ H(q)]
    Z
    H(q)
    = f(q)
    Order parameter Definition of Landau functional
    P(⇤) =
    ⇥ N
    i=1
    dqi⇥[⇤ f(q)]
    exp[ H(q)
    Z

    exp[ L]
    Z
    P(⇥) ⇥
    exp[ L]
    Z

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  13. Explicit DOF coarse-graining ...
    H(s1, s2, . . . , sN
    ) = J
    ij⇥
    sisj
    Order parameter.
    Block spin.
    Coarse-grained
    magnetization.
    =
    1
    N
    i
    si

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  14. Explicit DOF coarse-graining ...
    H(s1, s2, . . . , sN
    ) = J
    ij⇥
    sisj
    Order parameter.
    Block spin.
    Coarse-grained
    magnetization.
    =
    1
    N
    i
    si
    P(⇥ ) ⇥
    exp[ L]
    Z⇥
    =?
    K. Binder, Z. Phys B, 43, 119 (1981)

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  15. Explicit DOF coarse-graining ...
    H(s1, s2, . . . , sN
    ) = J
    ij⇥
    sisj
    Order parameter.
    Block spin.
    Coarse-grained
    magnetization.
    =
    1
    N
    i
    si
    P(⇥ ) ⇥
    exp[ L]
    Z⇥
    =?
    K. Binder, Z. Phys B, 43, 119 (1981)

    View Slide

  16. Explicit DOF coarse-graining ...
    H(s1, s2, . . . , sN
    ) = J
    ij⇥
    sisj
    Order parameter.
    Block spin.
    Coarse-grained
    magnetization.
    =
    1
    N
    i
    si
    P(⇥ ) ⇥
    exp[ L]
    Z⇥
    =?
    K. Binder, Z. Phys B, 43, 119 (1981)
    Explicit DOF needs this!

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  17. Or ansatz based on symmetry.
    L = A
    2
    2 + B
    4
    4 + K
    2
    (⇥ )2
    P(⇥) ⇥ exp[ (A
    2 ⇥2 + B
    4 ⇥4)]
    P(s) ⇥ a+
    (s 1) + a (s + 1) -J +J
    local part non-local part
    K( )2
    All equal-time equilibrium properties are determined by this functional.
    !! Why is this called a “free energy” ? Is it a thermodynamic free energy ?

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  18. What about dynamics ? Temporal coarse-graining
    D = kBT
    6⇥ a
    Why is this independent of
    particle mass ?
    ⇥d
    = m Inertial time scale. Time the
    velocity ‘remembers’ its
    initial condition.
    d
    ˙
    x = v
    ˙
    x = ⇥(t)
    Overdamped equations of
    motion. Valid when
    d

    v + v = ⇤(t)
    = 6⌅⇥a

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  19. Overdamped stochastic equations
    damping reversible force
    fluctuation
    deterministic stochastic
    mean behaviour ‘root N’ fluctuations
    ˙
    x = ⇥U + ⇥(t)

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  20. Overdamped coarse-grained equations of motion
    ˙
    ⇤(r) =
    L

    + ⇥(r, t)
    ˙
    ⇥(r) = [A⇥ + B⇥3 K⇥2⇥] + (r, t)
    damping reversible force
    fluctuation
    Model A equation
    ⇥⇥(r, t)⇥(r , t )⇤ = 2kBT (r r ) (t t )
    Fluctuation Dissipation
    Relation.
    !! Zero-mean
    Gaussian noise
    (r, t)⇥ = 0
    “Stochastic partial differential equations with additive Gaussian noise”

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  21. Conserved order parameters
    Ising spin Lattice gas
    Ising spins are usually non-conserved
    d
    dt
    (r, t) = 0
    Lattice gas models are conserved
    d
    dt
    (r, t) = 0

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  22. Conserved coarse-grained dynamics
    damping reversible force
    fluctuation
    Model B equation
    Fluctuation Dissipation
    Relation.
    !! Zero-mean vector
    Gaussian noise
    (r, t)⇥ = 0
    ˙
    ⇤(r) = ⇤2
    L

    + ⇤ · ⇥(r, t)
    ˙
    ⇥(r) = ⇤2[A⇥ + B⇥3 K⇤2⇥] + ⇤ · (r, t)
    !! Derive this. Use the local conservation law
    ˙ + ⇥ · j = 0
    ⇥⇥i
    (r, t)⇥j
    (r , t )⇤ = 2kBT ij
    (r r ) (t t )

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  23. Numerical solution
    Stochastic PDE
    Stochastic ODE
    Stochastic realisation
    Spatial discretisation
    Temporal integration
    ⇤t⇥ = D⇤2
    x
    ⇥ +
    ⇥x
    =
    (x + h) (x)
    h
    ⇤t⇥(i) = DL2
    ij
    ⇥(j) + (i)
    ⇥(i, t + t) ⇥(i, t) =
    t+
    t
    DL2
    ij
    ⇥(j) + (i)
    !! Obtain the explicit form of the L matrix for a central difference Laplacian

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  24. Summary of TDGL mesoscale methods
    Microscopic Hamiltonian
    Ginzburg-Landau Functional
    Langevin equations
    Explicit Coarse Graining
    Overamped approximation
    Symmetry + gradient expansion
    Numerical solution

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  25. Conclusion
    • Mesoscale methods are appropriate at length scales intermediate between the molecular
    and continuum scales.
    • The continuum description must be supplemented by fluctuation terms. At the mesoscale,
    the number of particles is not so large that fluctuations can be neglected.
    • Lengths and times must be coarse-grained. Intelligent coarse-graining improves
    computational efficiency, often by orders of magnitude, compared to direct MD.
    • Mesoscale methods can be particle-based, for instance dissipative particle dynamics and
    Brownian dynamics, or field-based like time-dependent Ginzburg-Landau theory and
    fluctuating hydrodynamics.
    • TDGL equations can be solved very efficiently on computers using matrix formulations.

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  26. Further reading
    • Stochastic processes in Physics and Chemistry : van Kampen
    • Handbook of stochastic processes : Gardiner
    • Principles of Condensed Matter Physics : Chaikin and Lubensky
    • Modern Theory of Critical Phenomena : Ma
    • Reviews of Modern Physics article : Halperin and Hohenberg
    • Numerical Solutions of Stochastic Differential Equations : Kloeden and Platen.

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  27. Thank you for your attention.

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